Nilpotent group

Nilpotent group is a term from the field of group theory, a branch of mathematics. In a sense, he generalized for finite groups the notion of commutative group " as little as possible ": Every commutative group is nilpotent, but not vice versa. Finite commutative groups can be represented ( up to isomorphism ) uniquely as a direct product of finitely many cyclic groups of prime power order. This is a statement of the fundamental theorem on finitely generated abelian groups. At finite nilpotent groups, the p- Sylowgruppen take on the role of cyclic groups: Every finite nilpotent group is ( up to isomorphism ) a direct product of its p- Sylowgruppen. The definition of " nilpotent group " is based on the more general concept of a chain of subgroups (with certain properties ), which is in the article " series ( group theory ) " below.

Characterizations

For nilpotent groups, various equivalent characterizations can be stated. They are often introduced through the study of particular rows. Define a group commutators for and receive the descending central series. Call nilpotent if the descending central series ends with one group for one.

Similarly, one can be by virtue of the -th center, and define it as the inverse image of in. This is an ascending series; the ascending central series. It then shows that exactly then is nilpotent in the above sense, if this series up to quite rises and that the lengths of the two strings are equal, which (also Nilpotenzgrad ) leads to the definition of nilpotency.

Further characterizations of nilpotent are:

  • All - Sylowuntergruppen are normal in. In particular, is a direct product of their - Sylowuntergruppen.
  • For primes products of elements are again elements.
  • Each subgroup is subnormal of.
  • For distinct primes and the commutators of elements elements are equal to the neutral element.
  • Is a proper subgroup of, then is genuine in its normalizer included.
  • Is a maximal subgroup, as is normal in.

Properties

  • Subgroups, factor groups and homomorphic images of a nilpotent group is nilpotent.
  • Every nilpotent group is solvable. ( The converse is false in general. )
  • Products nilpotent normal subgroup in a group are nilpotent. This property leads to the definition of the Fitting subgroup (after Hans Fitting) the product of all nilpotent normal subgroup.

Classification

  • The direct product of nilpotent groups is nilpotent if the Nilpotenzgrade the factors are limited.
  • Every finite p- group is nilpotent. An infinite p- group is nilpotent if the order of the group elements is limited. (Note that this requirement is stronger than the requirement of finite order for group elements which is guaranteed by the definition of the p- group anyway. )
  • A finite nilpotent group is isomorphic to the direct product of its p- Sylow subgroups. It should be noted here that every nilpotent group at any prime p exactly one (possibly trivial ) has p- Sylow subgroup.

Examples

  • A group is nilpotent if and only from Nilpotenzgrad 1 if it is abelian.
  • There is a body and a natural number. The amount of the n × n- matrices of the form
  • The dihedral group of elements is nilpotent if and only if the following applies; In this case, the Nilpotenzgrad equal.
  • The Frattinigruppe is always nilpotent and nilpotent if, then.
General linear group Frattini subgroup
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